Financial market traders and investors at commercial and investment banks, hedge funds, other managed funds, pension funds, corporate treasuries, insurance companies and elsewhere (including individual investors) have an interest in the use of options written on an underlying financial instrument for investment trading and hedging purposes. For example, a company might wish to hedge against future price movements in oil or a currency exchange rate, and an investment bank proprietary trader might wish to take a view (an investment position) on the future price movements of a stock. Accordingly, the accurate valuation of options is important to a wide range of financial market participants. Further, option prices convey information about expected future prices (e.g., future interest rates) that are of interest to other market observers, such as economic forecasters and central banks.